04198250 Social Networks 2015

Lecture 03: Networks in their surrounding contexts

The role of surrounding contexts

● Triadic closure: explains link formaton by propertes intrinsic in the network

● Today we will look at how factors outside the standard network model infuences link formaton

● In partcular: similarity between individuals● .. and the other way around

Homophily

• “Birds of a feather fock together.”• Your friends are more similar to you in age,

race, interests, opinions, etc. than a random collecton of individuals

• Homophily: principle that we tend to be similar to our friends

• One of the most basic notons governing the formaton of a social network

Moody (2001)

Social network from town’s middle and high schools

Moody (2001)

Social network from town’s middle and high schools

Clustering by race

Moody (2001)

Social network from town’s middle and high schools

Clustering by age

Homophily and Triadic Closure• Triadic closure: When two individuals share a

common friend, a friendship between them is more likely to occur

• Homophily suggests two individuals are more alike because of common friend, so link may occur even if neither is aware of mutual friend!

• Difcult to atribute formaton of link to any one factor

Can we develop a simple test for the

presence of homophily in a network?

Example Male/Female Network9 nodes: 6 males, 3 females

11 edges: 6 m-m, 2 f-f, 2 m-f

Does it exhibit homophily?

We have Homophily if few cross-edges

• Assume n nodes, with pn males and qn females (fractons p and q with 0<p,q<1 and p+q=1).

• If there is no homophily then edges are random.

If nodes randomly assigned a gender, then probability of an edge being male-male is p², female-female is q² and cross-gender is 2pq.

• Thus, if << 2pq cross-gender edges then the network shows homophily!

p²

q²2pq

Figure out p, q, n and decide if the network below exhibits homophily.

Notes on Homophily Test

• What if network had significantly more than 2pq cross-gender edges?

– Inverse homophily

– Example: Male-Female datng relatonships

Notes on Homophily Test

• Homophily test can be used to test any characteristc like race, age, natve language, preferences, etc.

• For characteristcs that have more than 2 values, perform same type of calculaton

• Compare number of heterogeneous edges to what randomly generated graph would look like using real data as probabilites

Why is homophilu ofen present in a social network?

• Two answers: Selecton and social infuence

• Selection: People tend to choose friends that are like themselves

• Can operate at different scales and levels of intentonality

– You actvely choose friends that are like yourself among a small group of people

– Your school’s populaton is relatvely homogeneous compared to overall populaton, so your environment compels you to choose friends like yourself

Mutable and Immutable Characteristcs

• Link formaton operates differently based on type of characteristc

• Immutable: Characteristcs that don’t change (gender, race) or change consistently with the populaton (age, generaton)

• Mutable: Characteristcs that can change over tme (behaviors, beliefs, interests, opinions)

Selecton and Social Infuence

• Research has shown that people are susceptble to social influence: they may change their behaviors to more closely resemble the behaviors of their friends (surprise!)

• “Bad company corrupts good character.” – 1 Corinthians 15:33

• Social infuence is reverse of selecton.

– Selecton: Individual characteristcs drive the formaton of links

– Social infuence: Existng links shape people’s mutable characteristcs

Longitudinal Studies

• Difcult to tell if selecton or social infuence at play with a single snapshot of a network

• Longitudinal studies tracking social connectons and individual behaviors over tme can help researchers uncover effect of social infuence

• Does behavior change afer changes to the network, or does the network change afer changes in behavior?

Example• Longitudinal studies have been used to determine if selecton

or social infuence has had more effect on teenagers’ scholastc achievement and drug use

• Understanding these effects can be helpful in developing interventons

• Assume drug use displays homophily in a network. Then, a

program to target social infuence (get friends to infuence other friends to stop) might work well, but...

• But if homophily is due to selecton, former drug users may choose new friends and drug-using behavior of others is not strongly affected

Social Infuence of Obesity

• Christakis and Fowlertracked obesity statusand social networkof 12,000 people over32 years

• Found homophily based on obesity status• They wanted to know why

Christakis & Fowler (2007)htp://www.ted.com/talks/nicholas_christakis_the_hidden_infuence_of_social_networks.html

Social Infuence of Obesity

• Christakis and Fowlertracked obesity statusand social networkof 12,000 people over32 years

• Found homophily based on obesity status• They wanted to know why

Christakis & Fowler (2007)htp://www.ted.com/talks/nicholas_christakis_the_hidden_infuence_of_social_networks.html

Why Obesity Homophily?

1. Selecton effects – people choose to befriend others of similar obesity status?

2. Confounding effects of homophily – other factors that correlate with obesity status?

3. Social infuence – if friends changed their obesity status, did it infuence person’s future obesity status?

• Discovered significant evidence for hypothesis 3: Obesity is a

type of “contagion” that can spread through social infuence!

Afliaton Network• Putng context into the network by showing connectons to

actvites, companies, organizatons, neighborhoods, etc.

• Bipartite graph: every edge joins two nodes belonging to different sets

Sue

Bill

Chess Club

Band

Social-Afliaton Network

• Two types of edges:– Person to person: Friendship or other social relatonship

– Person to foci: Partcipaton in the focus

Sue

Bill

Chess Club

Band

Gary

Alice

Closure Processes

SueGary

Alice

Triadic closure

Band Sue

Bill

Sue

Gary

Band

Focal closure: closure due to selecton

Membership closure: closure due to social infuence

Research Questons About Closure• Would Sue be more likely to become friends with Alice if they

shared more than one friend?

• In other words, is triadic closure dependent on the number of shared friends?

• More formally: What is the probability that two people form a link as a functon of the number of mutual friends they share?

SueGary

Alice

SueGary

Alice Moe

Research Questons About Closure

• For focal closure, what is the probability that two people form a link as a functon of the number of foci they are jointly afliated with?

• For membership closure, what is the probability that a person becomes involved in a partcular focus as a functon of the number of friends who are already involved in it?

Band Sue

Bill Ping Pong

Sue

Gary

Band

Jill

Research Methodology: Measuring Triadic Closure

1. Take two snapshots of network at tmes t1 and t2

2. For each k, identfy all pairs of nodes who have exactly k friends in common at t1 but who are not directly connected by an edge

3. Define T(k) to be fracton of these pairs that form an edge by

t2. This is the probability that a link will form between two people with k friends in common

4. Plot T(k) as functon of k to illustrate effect of common friends on link formaton

Email Social Network

• Kossinets and Wats (2006) examined email communicaton of 22,000 students over one year at a large US university

• Made link between two people if an email was sent between the two in the last 60 days

• Each snapshot is one day apart

• T(k) averaged over multple pairs of snapshots

Triadic Closure in Email Data Set

Almost no emails

exchanged when no friends in common

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Triadic Closure in Email Data Set

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Significant increase

going from 1 to 2 friends

Significant increase but

on much smaller

subpopulation

Triadic Closure in Email Data Set

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Baseline model: probability p of

forming link when k friends in commonTbase(k) = 1 – (1 –

p)^k

Tbase(k) = 1 – (1 –p)^(k-1)

Email Social Network• To evaluate focal closure, Kossinets and Wats

obtained class schedules for 22,000 students• Created social-afliaton network where

classes are foci• Determined probability of link formaton as

functon of number of shared foci

ENG 101 Sue

Bill HIST 202

Focal Closure in Email Data Set

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Sharing a class has

nearly same effect as sharing a

friend

Diminishing returns

Tbase(k) = 1 – (1 – p)^k

Measuring Membership Closure

• Backstrom et al. (2006) created social-afliaton network for blogging site LiveJournal

– Friendships designated by users in their profile– Foci are membership in user-defined communites

Sue

Gary

Jazz Comm

Jill

Membership Closure in LiveJournal Data Set

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Significant increase

going from 1 to 2

Diminishing returns

Measuring Membership Closure

• Crandall et al. (2008) created social-afliaton network for Wikipedia

– Node for editors maintaining a user talk page

– Link if two editors have communicated using talk page

– Foci are artcles edited by editors

Sue

Gary

Jazz Artcle

Jill

Membership Closure in Wikipedia Data Set

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Significant increase

going from 1 to 2

Diminishing returns

Selecton and Social Infuence

• Further examinaton of Wikipedia data set to see evidence of homophily produced by selecton and social infuence

• How do we measure similarity of two editors?

number of artcles edited by both editorsnumber of artcles edited by at least one editor

• Similar to neighborhood overlap of two editors in an afliaton network of editors and artcles

Selecton and Social Infuence

• We have homophily! Pairs of Wikipedia editors who have communicated are significantly more similar in behavior than editors who have never communicated

• Does homophily arise because…– editors are talking with editors who have edited

the same artcle? (selecton)– editors are led to the artcles by those they talk

to? (social infuence)

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

Segregaton

• Cites ofen divided into homogeneous neighborhoods based on ethnic and racial lines

Figure from htp://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch04.pdf

High density of African-

Americans

Low density of African-

Americans

Schelling Model of Segregaton

• Thomas Schelling introduced simple model in 1970s that shows how global paterns of self-chosen segregaton can emerge due to the effects of homophily at the local level, even if no individual wants segregated outcome.

• Populaton formed by two types of individuals called agents (X and O)

• The two types of agents represent immutable characteristc like race, ethnicity, country of origin, or natve language

X X OX O O OX X X XX O X X

O O

Agents placed randomly in grid

Agents are satisfied with their locaton in the grid if at least t of their neighbors are agents of the same type.

t = threshold

In each round, first identfy all unsatsfied agents

X X OX O O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Unsatsfied because only 2 neighbors

are X

X X OX O O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Unsatsfied because only 2 neighbors are X

X* X OX O O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Satsfied because 3 neighbors are O

X* X* OX O O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Satsfied because 4

neighbors are X

X* X* OX O O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Unsatsfied

X* X* OX O O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Satsfied

X* X* OX O* O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Unsatsfied

X* X* OX O* O OX X X XX O X X

O O

Assume t is 3 – all agents prefer to have at least 3 neighboring agents that are of the same type

Etc… Here’s the final result where unsatsfied agents marked with *

X* X* OX O* O O*X X X X

X* O* X XO* O*

Now all unsatsfied agents move, one by one, to a locaton where they are satsfied

O O OX O O OX X X X X

X X X XO*

Previously satisfied agent no

longer satisfied!

Note that the grid is now more segregated than before

Sometimes an agent can’t find a new location that will satisfy – leave alone or move to random location

Model Variatons

• Schedule agents to move in random order or in sweeping moton

• Agents move to nearest locaton that satsfies or random one

• Threshold could be percentage or vary among or within populaton groups

• World could wrap (lef edge meets up with right edge)

• Many other variatons, but results usually end up the same:

self-imposed segregation!

Schelling Simulators

• Check out a simulator:• Uri Wilensky’s NetLogo Segregaton model

htp://ccl.northwestern.edu/netlogo/models/Segregaton

Luke’s Schelling Model using t = 3

Randomly placed Afer 20 rounds

Conclusions

• Spatal segregaton takes place even though no single agent is actvely seeking it

• Schelling model is an example of how mutable characteristcs can become highly correlated with immutable characteristcs

• Choice of where to live (mutable) over tme conforms to agents’ type (immutable) producing segregaton (homophily)